A transport company has detailed records of all its deliveries. The number of minutes a delivery is made before or after its schedule delivery time can be modelled as a normally distributed random variable, `T`, with a mean of zero and a standard deviation of four minutes. A graph of the probability distribution of `T` is shown below.
- If `"Pr"(T <= a)=0.6`, find `a` to the nearest minute. (1 mark)
- Find the probability, correct to three decimal places, of a delivery being no later than three minutes after its scheduled delivery time, given that it arrives after its scheduled delivery time. (2 marks)
- Using the model described above, the transport company can make 46.48% of its deliveries over the interval `-3 <= t <= 2`.
- It has an improved delivery model with a mean of `k` and a standard deviation of four minutes.
- Find the values of `k`, correct to one decimal place, so that 46.48% of the transport company's deliveries can be made over the interval `-4.5 <= t <= 0.5` (3 marks)
A rival transport company claims that there is a 0.85 probability that each delivery it makes will arrive on time or earlier.
Assume that whether each delivery is on time or earlier is independent of other deliveries.
- Assuming that the rival company's claim is true, find the probability that on a day in which the rival company makes eight deliveries, fewer than half of them arrive on time or earlier. Give your answer correct to three decimal places. (2 marks)
- Assuming that the rival company's claim is true, consider a day in which it makes `n` deliveries.
- i. Express, in terms of `n`, the probability that one or more deliveries will not arrive on time or earlier. (1 mark)
- ii. Hence, or otherwise, find the minimum value of `n` such that there is at least a 0.95 probability that one or more deliveries will not arrive on time or earlier. (1 mark)
- An analyst from a government department believes the rival transport company's claim is only true for deliveries made before 4 pm. For deliveries made after 4 pm, the analyst believes the probability of a delivery arriving on time or earlier is `x`, where `0.3 <=x <= 0.7`
- After observing a large number of the rival transport company's deliveries, the analyst believes that the overall probability that a delivery arrives on time or earlier is actually 0.75
- Let the probability that a delivery is made after 4 pm be `y`.
- Assuming that the analyst's belief are true, find the minimum and maximum values of `y`. (2 marks)